When considering the factor model, we aim at finding a structured matrix \({\Sigma }\)  in \(\mathcal {M}_{p,k}\) , which is a smooth submanifold of \(\mathcal {S}^{++}_p\) .
The Riemannian geometry of \(\mathcal {M}_{p,k}\)  is not straightforward.
In fact, defining an adequate geometry for low-rank matrices is a rather difficult task and, while there were many attempts, see e.g., [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, no perfect solution has been found yet.
Here, we choose the parametrization considered in [1]}, [2]}, [6]}.
From a geometrical perspective, the novelty here is to adopt the particular structure of \(\mathcal {M}_{p,k}\) , i.e., by combining a low-rank positive semi-definite matrix and a diagonal positive definite matrix.
Each of the two manifolds of this product are well-known and the proofs of the following can easily be deduced from the proofs in [6]}.
All \({\Sigma }\in \mathcal {M}_{p,k}\)  can be written as
\({\Sigma } = {V}{\Lambda }{V}^{\top } + {\Psi },\) 
